Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #525×C79 ◁ G | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 127 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,32)(21,25)(22,27)(24,34)(26,36)(28,43)(29,38)(30,39)(31,41)(33,40)(35,42)(37,44)(45,47)(46,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,30)(18,23,27,39)(20,28,35,45)(21,31,36,29)(24,37,40,48)(25,41,26,38)(32,43,42,47)(33,46,34,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,28)(21,33)(22,35)(23,37)(25,32)(26,42)(27,40)(29,44)(30,45)(31,46)(34,36)(38,47)(39,48)(41,43);(3,11)(6,9)(10,16)(13,15)(17,21,22,36)(18,25,27,26)(19,29,30,31)(20,33,35,34)(23,38,39,41)(24,32,40,42)(28,44,45,46)(37,47,48,43);(2,8)(6,9)(7,14)(13,15)(19,30)(23,39)(28,45)(29,31)(37,48)(38,41)(43,47)(44,46)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54757);